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Yayın The boundary layer approximation and nonlinear waves in elastic tubes(Pergamon-Elsevier Science, 2000-09) Antar, Nalan; Demiray, HilmiIn the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and approximate equations of an incompressible viscous fluid, the propagation of weakly nonlinear waves is examined. In order to include the geometrical and structural dispersion into analysis, the wall's inertial and shear deformation are taken into account in determining the inner pressure-inner cross sectional area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves, in the long-wave approximation, are shown to be governed by the Korteweg-de Vries (KdV) and the Korteweg-de Vries-Burgers (KdVB), depending on the balance between the nonlinearity, dispersion and/or dissipation. In the case of small viscosity (or large Reynolds number), the behaviour of viscous fluid is quite close to that ideal fluid and viscous effects are confined to a very thin layer near the boundary. In this case, using the boundary layer approximation we obtain the viscous-Korteweg-de Vries and viscous-Burgers equations.Yayın Weakly nonlinear waves in a fluid with variable viscosity contained in a prestressed thin elastic tube(Pergamon-Elsevier Science Ltd, 2008-04) Demiray, HilmiIn this work, treating the artery as a prestressed thin elastic tube and the blood as an incompressible Newtonian fluid with variable viscosity which vanishes on the boundary of the tube, the propagation of nonlinear waves in such a fluid-filled elastic tube is studied, in the longwave approximation, through the use of reductive perturbation method and the evolution equation is obtained as the Korteweg-deVries-Burgers equation. A progressive wave type of solution is presented for this evolution equation and the result is discussed.Yayın Solitary waves in elastic tubes filled with a layered fluid(Pergamon-Elsevier Science, 2001-04) Demiray, HilmiIn this work, we studied the propagation of weakly non-linear waves in a prestressed thin elastic tube filled with an incompressible layered fluid, where the outer layer is assumed to be inviscid whereas the cylindrical core is considered to be viscous. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is studied. The governing equation is shown to be the perturbed Korteweg-de Vries (KdV) equation. A travelling wave type of solution for this evolution equation is sought and it is shown that the amplitude of the solitary wave for the perturbed KdV equation decays slowly with time.Yayın A factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid(Elsevier Science inc, 2005-05-25) Tunga, Mehmet Alper; Demiralp, MetinWhen the values of a multivariate function f(x(1),...,x(N)), having N independent variables like x(1),...,x(N) are given at the nodes of a cartesian, product set in the space of the independent variables and ail interpolation problem is defined to find out the analytical structure of this function some difficulties arise in the standard methods due to the multidimensionality of the problem. Here, the main purpose is to partition this multivariate data into low-variate data and to obtain the analytical structure of the multivariate function by using this partitioned data. High dimensional model representation (HDMR) is used for these types of problems. However, if HDMR requires all components, which means 2(N) number of components, to get a desired accuracy then factorized high dimensional model representation (FHDMR) can be used. This method uses the components of HDMR. This representation is needed when the sought multivariate function has a multiplicative nature. In this work we introduce how to utilize FHDMR for these problems and present illustrative examples.Yayın Hybrid high dimensional model representation (HHDMR) on the partitioned data(Elsevier B.V., 2006-01-01) Tunga, Mehmet Alper; Demiralp, MetinA multivariate interpolation problem is generally constructed for appropriate determination of a multivariate function whose values are given at a finite number of nodes of a multivariate grid. One way to construct the solution of this problem is to partition the given multivariate data into low-variate data. High dimensional model representation (HDMR) and generalized high dimensional model representation (GHDMR) methods are used to make this partitioning. Using the components of the HDMR or the GHDMR expansions the multivariate data can be partitioned. When a cartesian product set in the space of the independent variables is given, the HDMR expansion is used. On the other band, if the nodes are the elements of a random discrete data the GHDMR expansion is used instead of HDMR. These two expansions work well for the multivariate data that have the additive nature. If the data have multiplicative nature then factorized high dimensional model representation (FHDMR) is used. But in most cases the nature of the given multivariate data and the sought multivariate function have neither additive nor multiplicative nature. They have a hybrid nature. So, a new method is developed to obtain better results and it is called hybrid high dimensional model representation (HHDMR). This new expansion includes both the HDMR (or GHDMR) and the FHDMR expansions through a hybridity parameter. In this work, the general structure of this hybrid expansion is given. It has tried to obtain the best value for the hybridity parameter. According to this value the analytical structure of the sought multivariate function can be determined via HHDMR.Yayın Weakly nonlinear waves in a prestressed thin elastic tube containing a viscous fluid(Pergamon-Elsevier Science Ltd, 1999-11) Antar, Nalan; Demiray, HilmiIn this work, we studied the propagation of weakly nonlinear waves in a prestressed thin elastic tube filled with an incompressible viscous fluid. In order to include the geometrical and structural dispersion into analysis, the wall's inertial and sheer deformation are taken into account in determining the inner pressure-inner cross sectional area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves, in the long-wave approximation, is shown to be governed by the Korteweg-de Vries-Burgers (KdVB) equation. Due to dependence of coefficients of the governing equation on the initial deformation, the material and viscosity parameters, the profile of the travelling wave solution to the KdVB equation changes with these parameters. These variations are calculated numerically for some elastic materials and the effects of initial deformation and the viscosity parameter on the propagation characteristics are discussed.Yayın Weakly nonlinear waves in a viscous fluid contained in a viscoelastic tube with variable cross-section(Gauthier-Villars/Editions Elsevier, 2005-03) Demiray, HilmiIn the present work, treating the arteries as a thin walled prestressed viscoelastic tube with variable cross-section, and using the longwave approximation, we have studied the propagation of weakly nonlinear waves in such a fluid-filled viscoelastic tube by employing the reductive perturbation method. By considering the blood as an incompressible viscous fluid, depending on the order of various physical entities, various evolution equations with variable coefficients are obtained and progressive wave solutions to these evolution equations are given whenever possible. It is shown that this type of equations admit solitary wave type of solutions with variable wave speeds.Yayın An analytical solution to the dissipative nonlinear Schrodinger equation(Elsevier Science, 2003-12-20) Demiray, HilmiMotivated with a solitary wave type of solution to the nonlinear Schrodinger (NLS) equation, in this work we shall seek a travelling wave solution to the dissipative NLS equation by use of the hyperbolic tangent method. It is observed that the dissipative NLS equation still assumes a solitary wave type of solution with decaying amplitude in the time parameter.Yayın Head-on collision of solitary waves in fluid-filled elastic tubes(Pergamon-Elsevier Science Ltd, 2005-08) Demiray, HilmiIn this work, treating the arteries as a thin walled, prestressed thin elastic tube and the blood as an inviscid fluid, we have studied the propagation of nonlinear waves, in the longwave approximation, through the use of extended PLK perturbation method. The results show that, up to O(epsilon(2)), the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the collision. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly.Yayın Waves in fluid-filled elastic tubes with a stenosis: Variable coefficients KdV equations(Elsevier B.V., 2007-05-15) Demiray, HilmiIn the present work, by treating the arteries as thin-walled prestressed elastic tubes with a stenosis and the blood as an inviscid fluid we have studied the propagation of weakly nonlinear waves in such a medium, in the longwave approximation, by employing the reductive perturbation method. The variable coefficients KdV and modified KdV equations are obtained depending on the balance between the nonlinearity and the dispersion. By seeking a localized progressive wave type of solution to these evolution equations, we observed that the wave speeds takes their maximum values at the center of stenosis and gets smaller and smaller as one goes away from the stenosis. Such a result seems to reasonable from the physical point of view.
